Presentation Record for Paper Number 75l
A New Approach for Reliable Computation of
Homogeneous Azeotropes in Multicomponent Mixtures
Robert W. Maier, Joan F. Brennecke and Mark A. Stadtherr
Department of Chemical Engineering
University of Notre Dame, Notre Dame, IN 46556
Prepared for presentation at AIChE Annual Meeting, Los Angeles, CA, November 16{21, 1997
Session Number 75: Thermodynamic Properties and Phase Behavior : General II
c Robert W. Maier
Copyright UNPUBLISHED
October 1997
AIChE shall not be responsible for statements or opinions contained in papers or printed in its
publications.
author to whom all correspondence should be addressed; Fax: (219) 631-8366; E-mail: [email protected]
1 Introduction
The determination of the existence and composition of azeotropes is important both
from theoretical and practical standpoints. An important test of the veracity of thermodynamic models is whether or not known azeotropes are predicted, and whether or not
they are predicted accurately. Model parameters can be ne tuned by comparing the model
predictions with known azeotropic data. In addition, azeotropic predictions can be used
as starting points for experimental searches for actual azeotropes. These azeotropes often
present limitations in process design which must be known, and their determination strictly
from experiment alone can be expensive.
There are two main diculties in solving the problem. The rst is the fact that the
equations derived from most thermodynamic models are highly nonlinear, which may make
nding any azeotrope a nontrivial exercise. In addition, there is the question of whether or
not all of the azeotropes have been found, or of being certain that there are no azeotropes
if none have been found. Because of these diculties there has been much recent interest in
the reliable computation of azeotropes, focused primarily on the prediction of homogeneous
azeotropes. For example, Fidkowski et al. (1993) have presented a homotopy continuation
method for the calculation of homogeneous azeotropes. The primary drawback of this technique is that it can not guarantee that all azeotropes have been found. More recently, Harding
et al. (1997) have reported a global optimization procedure based on a branch and bound
approach using convex underestimating functions that are continuously rened as the range
where azeotropes are possible is narrowed. This technique does provide a guarantee that
all azeotropes have been found. Harding et al. (1997) have developed appropriate convex
underestimating functions for several specic thermodynamic models.
2 Methodology
We describe here a new approach for reliably nding all homogeneous azeotropes of
multicomponent mixtures. The technique is based on interval analysis, in particular the
use of an interval Newton/generalized bisection algorithm. The method can determine with
mathematical certainty all azeotropes for any system. The technique is general-purpose and
can be applied in connection with any thermodynamic models. No model-specic convex
underestimating functions need be derived. In this presentation, the technique is described
and tested and some of the results are discussed.
2.1 Problem Formulation
The equilibrium condition can be rewritten in terms of the equivalence of fugacities,
f^iV = f^iL ; 8i 2 N
(1)
where f^i refers to the fugacity of component i in solution, and N is the set of all components
2
in the system. These solution fugacities can in turn be written as
yi^Vi P = xi iL fiL ; 8i 2 N
(2)
where ^Vi is the mixture fugacity coecient of component i in the vapor phase, iL is the
activity coecient of component i in the liquid phase, and fiL is the pure component fugacity
of component i in the liquid phase. The pure component fugacity in the liquid phase can be
expressed as
sat
fiL = sat
(3)
i Pi (P F )i ; 8i 2 N
sat
where sat
i is the saturated pure component fugacity component for species i, Pi is the
vapor pressure of component i, and (P F )i is the Poynting correction factor for species i.
Under the assumption of ideal gas behavior of the vapor phase,
^Vi = sat
(4)
i = (P F )i = 1; 8i 2 N
Noting that the mole fraction of each component is the same in both the liquid and vapor
phases for homogeneous azeotropes, we arrive at the following equilibrium conditions for
homogeneous azeotropy,
= PisatiL; 8i 2 N
(5)
In addition to the equilibrium equations, we also have the constraint that the sum of the
mole fractions must equal one.
P
N
X
i=1
xi , 1 = 0
(6)
Since the vapor pressure and activity coecient relationships are expressed in terms of their
logarithms, we express the set of equations (5) as
ln P , ln Pisat , ln iL = 0; 8i 2 N
(7)
In the terminology of Harding et al. (1997), this is the N-ary formulation, in which each
of the N components must have a non-zero mole fraction. In order to nd all azeotropes of
a system using the N-ary formulation, a series of problems must be run, covering each of
the possible component combinations. The k-ary formulation, which allows for zero mole
fractions, is derived by multiplying the equilibrium conditions (7) by xi to obtain
ln P , ln Pisat , ln iL = 0; 8i 2 N
(8)
Using the k-ary formulation, only one problem need be run to nd all azeotropes present in
a system.
In this paper, we use the Antoine equation to determine pure component vapor pressures,
and present results for an example in which the Wilson activity coecient model is used.
Problems involving other models, such as NRTL, UNIQUAC and UNIFAC, have also been
solved.
xi
3
2.2 Interval Newton/Generalized Bisection
The solution method used is the interval Newton/generalized bisection technique described by Kearfott (1987a,b), and implemented in INTBIS (Kearfott and Novoa, 1990).
The algorithm is also summarized, in the context of chemical process modeling, by Schnepper and Stadtherr (1996).
For a system of nonlinear equations f (x) = 0 with x 2 X , the basic iteration step
in interval Newton methods is, given an interval X k , to solve the linear interval equation
system
F 0 (X k )(N k , x k ) = ,f (x k )
(9)
for a new interval N k , where k is an iteration counter, F 0(X k ) is an interval extension
of the real Jacobian f 0(x) of f (x) over the current interval X k , and x k is a point in the
interior of X k . It can be shown (Moore, 1966) that any root x 2 X k is also contained in
the image N k , suggesting the iteration scheme X k = X k \ N k . While this iteration
scheme can be used to tightly enclose a solution, what is also of signicance here is the
power of equation (9) as an existence and uniqueness test. For several techniques for nding
N k from equation (9), it can be proven (e.g., Neumaier, 1990; Kearfott, 1996) that if N k
X k , then there is a unique zero of f (x) in X k , and that Newton's method with real
arithmetic can be used to nd it, starting from any point in X k . This suggests a root
inclusion test for X k :
1. (Range Test) Compute an interval extension F(X k ) containing the range of f (x) over
X k and test to see whether it contains zero. Clearly, if 0 2= F(X k ) ff (x) j x 2
X k g then there can be no solution of f (x) = 0 in X k and this interval need not be
further tested.
2. (Interval Newton Test) Compute the image N k by solving equation (9).
(a) If X k \ N k = ;, then there is no root in X k .
(b) If N k X k , then there is exactly one root in X k .
(c) If neither of the above is true, then no further conclusion can be drawn.
In the last case, one could then repeat the root inclusion test on the next interval Newton
iterate X k , assuming it is suciently smaller than X k , or one could bisect X k and
repeat the root inclusion test on the resulting intervals. This is the basic idea of interval
Newton/generalized bisection methods. If f (x) = 0 has a nite number of real solutions
in the specied initial box, a properly implemented interval Newton/generalized bisection
method can nd with mathematical certainty any and all solutions to a specied tolerance,
or can determine with mathematical certainty that there are no solutions in the given box
(Kearfott and Novoa, 1990; Kearfott, 1990).
(0)
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( +1)
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4
( +1)
3 Results and Discussion
We discuss the results from one of our test problems below. It is a three component
system consisting of ethanol, methyl ethyl ketone, and water. The activity coecient model
used is the Wilson equation. This system has also recently been studied by Fidkowski et
al. (1993) and Harding et al. (1997) We compare the results between the N-ary and k-ary
formulations for this system for two cases. In the rst case, the binary interaction parameters
are considered to be constant by assuming a reference temperature as done by Harding et
al. (1997), and in the second case, we solve the more dicult problem in which the binary
interaction parameters are not assumed to be constant and allowed to vary with temperature.
Our results for this problem are shown in Table 1 for the constant reference temperature
case and in Table 2 for the temperature dependent case. CPU times are for a Sun Ultra
1/140. Times for each azeotrope are shown for the N-ary formulation, and the total times
are shown for both formulations. In each case, both formulations give the same results for
the compositions and temperatures of the azeotropes found.
The results we obtained for the constant reference temperature case are in good agreement with those published by Harding et al. (1997) for the same test problem, indicating
that the method can eciently and reliably determine all homogeneous azeotropes for multicomponent mixtures. In each case, we obtained faster solution times for the series of
N-ary problems than for the single k-ary problem. The times required for the interval Newton/generalized bisection method are very competitive with those for the branch and bound
method of Harding et al. (1997). The computational cost of including temperature dependence is signicant, but the CPU times are still reasonable. Including the temperature
dependence is useful for systems with no experimental data, because it is dicult to choose
a reference temperature, and because for some problems, especially ternary and higher order
systems, the calculation of azeotropes is sensitive to the reference temperature.
4 References
Fidkowski, Z. T., M. F. Malone and M. F. Doherty, Computing azeotropes in multicomponent mixtures. Comput. Chem. Eng., 17, 1141-1155 (1993).
Harding, S. T., C. D. Maranas, C. M. McDonald and C. A. Floudas, Locating all homogeneous azeotropes in multicomponent mixtures. Ind. Eng. Chem. Res., 36, 160-178 (1997).
Kearfott, R. B., Abstract generalized bisection and a cost bound. Math. Comput., 49, 187202 (1987a).
Kearfott, R. B., Some tests of generalized bisection. ACM Trans. Math. Softw., 13, 197-220
(1987b).
5
Kearfott, R. B., Interval arithmetic techniques in the computational solution of nonlinear
systems of equations: Introduction, examples, and comparisons. Lectures in Applied Mathematics, 26, 337-357 (1990).
Kearfott, R. B., Rigorous Global Search: Continuous Problems. Kluwer Academic Publishers, Dordrecht (1996).
Kearfott, R. B., and Manuel Novoa III. Algorithm 681: INTBIS, a portable interval-Newton
/bisection package. ACM Transactions on Mathematical Software, 16(2):152-157, June 1990.
Moore, R. E., Interval Analysis. Prentice-Hall, Englewood Clis (1966).
Neumaier, A., Interval Methods for Systems of Equations. Cambridge University Press,
Cambridge, England (1990).
Schnepper, C. A. and M. A. Stadtherr, Robust process simulation using interval methods.
Comput. Chem. Eng., 20, 187-199 (1996).
6
Table 1: Computational results for the system ethanol(1)/methyl ethyl ketone(2)/water(3) at
1.0 atm for the Wilson equation with a reference temperature of 73.65 C used to determine
constant values for the binary interaction parameters. Temperature range for search was
10-100 C.
x
x
x
T ( C) CPU (s)
0.485 0.515 0.000 74.099 0.008
0.898 0.000 0.102 78.123 0.008
0.000 0.681 0.319 73.697 0.021
0.232 0.543 0.225 72.776 0.168
Total N-ary
0.205
Total k-ary
0.896
1
2
3
Table 2: Computational results for the system ethanol(1)/methyl ethyl ketone(2)/water(3) at
1.0 atm for the Wilson equation with temperature dependent binary interaction parameters.
Temperature range for search was 10-100 C.
x1
0.485
0.910
0.000
0.231
Total
Total
x2
0.515
0.000
0.681
0.544
N-ary
k-ary
x3
0.000
0.090
0.319
0.225
7
T ( C) CPU (s)
74.104 0.039
78.169 0.038
73.700 0.090
72.747 0.168
0.335
4.615